Find the point of inflection for f(x) = x^3 - 6x^2 + 9x. (2022)

Find the point of inflection for f(x) = x^3 - 6x^2 + 9x. (2022)

Step 1: Start with the function f(x) = x^3 - 6x^2 + 9x.
Step 2: Find the first derivative f'(x) by differentiating f(x).
Step 3: The first derivative is f'(x) = 3x^2 - 12x + 9.
Step 4: Find the second derivative f''(x) by differentiating f'(x).
Step 5: The second derivative is f''(x) = 6x - 12.
Step 6: Set the second derivative equal to zero: 6x - 12 = 0.
Step 7: Solve for x: 6x = 12, so x = 2.
Step 8: Find the y-coordinate by substituting x = 2 back into the original function: f(2) = 2^3 - 6(2^2) + 9(2).
Step 9: Calculate f(2): f(2) = 8 - 24 + 18 = 2.
Step 10: The point of inflection is (2, 2).

Second Derivative Test – The question tests the understanding of finding points of inflection using the second derivative of a function.
Critical Points – Identifying where the second derivative equals zero to find potential points of inflection.
Function Evaluation – Evaluating the original function at the point of inflection to find the corresponding y-coordinate.